Singular Value Consulting

Quintic root

Here’s a curious result I ran across the other day. Suppose you have a quintic equation of the form zx5 – x – 1 = 0. (It’s possible to reduce a general quintic equation to this form, known as Bring-Jerrard normal form.) There is no elementary formula for the roots of this equation, but the following infinite series does give a root as a function of the leading coefficient z:

One reason this is interesting is that the series above has a special form that makes it a hypergeometric function of z. You can read more about it here.

I could imagine situations where having such an expression for a root is useful, though I doubt the series would be much use if you just wanted to find the roots of a fifth degree polynomial numerically. Direct application of something like Newton’s method would be much simpler.

Just to follow up on series convergence… at first glance it does look like convergence is an issue. Playing with Stirling’s approximation you can show that asymptotically (5n-choose-n) is
sqrt(5/(8*pi*n))* (3125/256)^n. So one would expect convergence as long as |z| < (256/3125) == 0.08192.

A bit late, and of little importance: I believe the first sentence of second paragraph should read “One reason this is interesting is that the series above has a special form that makes it a hypergeometric function of z.” Replacing “is” => “it”.